# Properties

 Label 384.10.d.a.193.4 Level $384$ Weight $10$ Character 384.193 Analytic conductor $197.774$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$197.773761087$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225$$ x^8 + 13062*x^6 + 45211107*x^4 + 45928424926*x^2 + 852972309225 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{32}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 193.4 Root $$-90.8862i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.10.d.a.193.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-81.0000i q^{3} +2178.24i q^{5} -7658.16 q^{7} -6561.00 q^{9} +O(q^{10})$$ $$q-81.0000i q^{3} +2178.24i q^{5} -7658.16 q^{7} -6561.00 q^{9} -61495.3i q^{11} -28283.5i q^{13} +176438. q^{15} +193168. q^{17} +366846. i q^{19} +620311. i q^{21} -820746. q^{23} -2.79162e6 q^{25} +531441. i q^{27} +4.44100e6i q^{29} +2.59982e6 q^{31} -4.98112e6 q^{33} -1.66813e7i q^{35} -2.14501e7i q^{37} -2.29096e6 q^{39} -3.15678e7 q^{41} -9.42768e6i q^{43} -1.42915e7i q^{45} +3.59591e7 q^{47} +1.82938e7 q^{49} -1.56466e7i q^{51} -3.97195e7i q^{53} +1.33952e8 q^{55} +2.97145e7 q^{57} -9.36289e7i q^{59} +3.86390e7i q^{61} +5.02452e7 q^{63} +6.16083e7 q^{65} -1.80133e8i q^{67} +6.64805e7i q^{69} -1.54189e8 q^{71} -3.79531e8 q^{73} +2.26121e8i q^{75} +4.70940e8i q^{77} +5.21268e8 q^{79} +4.30467e7 q^{81} -5.93034e8i q^{83} +4.20768e8i q^{85} +3.59721e8 q^{87} +1.05828e8 q^{89} +2.16599e8i q^{91} -2.10585e8i q^{93} -7.99080e8 q^{95} +2.91252e8 q^{97} +4.03470e8i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 13632 q^{7} - 52488 q^{9}+O(q^{10})$$ 8 * q - 13632 * q^7 - 52488 * q^9 $$8 q - 13632 q^{7} - 52488 q^{9} - 90720 q^{15} + 8304 q^{17} - 4612608 q^{23} - 4754904 q^{25} - 7499328 q^{31} - 6213024 q^{33} - 23211360 q^{39} - 43518896 q^{41} + 49382016 q^{47} - 74106808 q^{49} + 19030656 q^{55} + 38141280 q^{57} + 89439552 q^{63} - 110270336 q^{65} - 741751296 q^{71} - 1507903440 q^{73} + 1008373440 q^{79} + 344373768 q^{81} + 423468000 q^{87} - 1337034448 q^{89} - 543950208 q^{95} - 904817936 q^{97}+O(q^{100})$$ 8 * q - 13632 * q^7 - 52488 * q^9 - 90720 * q^15 + 8304 * q^17 - 4612608 * q^23 - 4754904 * q^25 - 7499328 * q^31 - 6213024 * q^33 - 23211360 * q^39 - 43518896 * q^41 + 49382016 * q^47 - 74106808 * q^49 + 19030656 * q^55 + 38141280 * q^57 + 89439552 * q^63 - 110270336 * q^65 - 741751296 * q^71 - 1507903440 * q^73 + 1008373440 * q^79 + 344373768 * q^81 + 423468000 * q^87 - 1337034448 * q^89 - 543950208 * q^95 - 904817936 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 81.0000i − 0.577350i
$$4$$ 0 0
$$5$$ 2178.24i 1.55862i 0.626636 + 0.779312i $$0.284431\pi$$
−0.626636 + 0.779312i $$0.715569\pi$$
$$6$$ 0 0
$$7$$ −7658.16 −1.20554 −0.602772 0.797913i $$-0.705937\pi$$
−0.602772 + 0.797913i $$0.705937\pi$$
$$8$$ 0 0
$$9$$ −6561.00 −0.333333
$$10$$ 0 0
$$11$$ − 61495.3i − 1.26641i −0.773984 0.633205i $$-0.781739\pi$$
0.773984 0.633205i $$-0.218261\pi$$
$$12$$ 0 0
$$13$$ − 28283.5i − 0.274655i −0.990526 0.137328i $$-0.956149\pi$$
0.990526 0.137328i $$-0.0438513\pi$$
$$14$$ 0 0
$$15$$ 176438. 0.899872
$$16$$ 0 0
$$17$$ 193168. 0.560939 0.280470 0.959863i $$-0.409510\pi$$
0.280470 + 0.959863i $$0.409510\pi$$
$$18$$ 0 0
$$19$$ 366846.i 0.645792i 0.946434 + 0.322896i $$0.104656\pi$$
−0.946434 + 0.322896i $$0.895344\pi$$
$$20$$ 0 0
$$21$$ 620311.i 0.696021i
$$22$$ 0 0
$$23$$ −820746. −0.611553 −0.305776 0.952103i $$-0.598916\pi$$
−0.305776 + 0.952103i $$0.598916\pi$$
$$24$$ 0 0
$$25$$ −2.79162e6 −1.42931
$$26$$ 0 0
$$27$$ 531441.i 0.192450i
$$28$$ 0 0
$$29$$ 4.44100e6i 1.16598i 0.812481 + 0.582988i $$0.198116\pi$$
−0.812481 + 0.582988i $$0.801884\pi$$
$$30$$ 0 0
$$31$$ 2.59982e6 0.505609 0.252805 0.967517i $$-0.418647\pi$$
0.252805 + 0.967517i $$0.418647\pi$$
$$32$$ 0 0
$$33$$ −4.98112e6 −0.731163
$$34$$ 0 0
$$35$$ − 1.66813e7i − 1.87899i
$$36$$ 0 0
$$37$$ − 2.14501e7i − 1.88157i −0.338999 0.940787i $$-0.610088\pi$$
0.338999 0.940787i $$-0.389912\pi$$
$$38$$ 0 0
$$39$$ −2.29096e6 −0.158572
$$40$$ 0 0
$$41$$ −3.15678e7 −1.74468 −0.872341 0.488898i $$-0.837399\pi$$
−0.872341 + 0.488898i $$0.837399\pi$$
$$42$$ 0 0
$$43$$ − 9.42768e6i − 0.420530i −0.977644 0.210265i $$-0.932567\pi$$
0.977644 0.210265i $$-0.0674327\pi$$
$$44$$ 0 0
$$45$$ − 1.42915e7i − 0.519541i
$$46$$ 0 0
$$47$$ 3.59591e7 1.07490 0.537450 0.843295i $$-0.319388\pi$$
0.537450 + 0.843295i $$0.319388\pi$$
$$48$$ 0 0
$$49$$ 1.82938e7 0.453337
$$50$$ 0 0
$$51$$ − 1.56466e7i − 0.323859i
$$52$$ 0 0
$$53$$ − 3.97195e7i − 0.691453i −0.938335 0.345726i $$-0.887633\pi$$
0.938335 0.345726i $$-0.112367\pi$$
$$54$$ 0 0
$$55$$ 1.33952e8 1.97386
$$56$$ 0 0
$$57$$ 2.97145e7 0.372848
$$58$$ 0 0
$$59$$ − 9.36289e7i − 1.00595i −0.864301 0.502975i $$-0.832239\pi$$
0.864301 0.502975i $$-0.167761\pi$$
$$60$$ 0 0
$$61$$ 3.86390e7i 0.357308i 0.983912 + 0.178654i $$0.0571742\pi$$
−0.983912 + 0.178654i $$0.942826\pi$$
$$62$$ 0 0
$$63$$ 5.02452e7 0.401848
$$64$$ 0 0
$$65$$ 6.16083e7 0.428084
$$66$$ 0 0
$$67$$ − 1.80133e8i − 1.09209i −0.837757 0.546043i $$-0.816134\pi$$
0.837757 0.546043i $$-0.183866\pi$$
$$68$$ 0 0
$$69$$ 6.64805e7i 0.353080i
$$70$$ 0 0
$$71$$ −1.54189e8 −0.720097 −0.360049 0.932934i $$-0.617240\pi$$
−0.360049 + 0.932934i $$0.617240\pi$$
$$72$$ 0 0
$$73$$ −3.79531e8 −1.56421 −0.782105 0.623147i $$-0.785854\pi$$
−0.782105 + 0.623147i $$0.785854\pi$$
$$74$$ 0 0
$$75$$ 2.26121e8i 0.825212i
$$76$$ 0 0
$$77$$ 4.70940e8i 1.52671i
$$78$$ 0 0
$$79$$ 5.21268e8 1.50570 0.752851 0.658190i $$-0.228678\pi$$
0.752851 + 0.658190i $$0.228678\pi$$
$$80$$ 0 0
$$81$$ 4.30467e7 0.111111
$$82$$ 0 0
$$83$$ − 5.93034e8i − 1.37160i −0.727790 0.685801i $$-0.759452\pi$$
0.727790 0.685801i $$-0.240548\pi$$
$$84$$ 0 0
$$85$$ 4.20768e8i 0.874294i
$$86$$ 0 0
$$87$$ 3.59721e8 0.673176
$$88$$ 0 0
$$89$$ 1.05828e8 0.178790 0.0893952 0.995996i $$-0.471507\pi$$
0.0893952 + 0.995996i $$0.471507\pi$$
$$90$$ 0 0
$$91$$ 2.16599e8i 0.331109i
$$92$$ 0 0
$$93$$ − 2.10585e8i − 0.291914i
$$94$$ 0 0
$$95$$ −7.99080e8 −1.00655
$$96$$ 0 0
$$97$$ 2.91252e8 0.334038 0.167019 0.985954i $$-0.446586\pi$$
0.167019 + 0.985954i $$0.446586\pi$$
$$98$$ 0 0
$$99$$ 4.03470e8i 0.422137i
$$100$$ 0 0
$$101$$ 1.38250e9i 1.32197i 0.750401 + 0.660983i $$0.229860\pi$$
−0.750401 + 0.660983i $$0.770140\pi$$
$$102$$ 0 0
$$103$$ −5.22223e8 −0.457181 −0.228591 0.973523i $$-0.573412\pi$$
−0.228591 + 0.973523i $$0.573412\pi$$
$$104$$ 0 0
$$105$$ −1.35119e9 −1.08484
$$106$$ 0 0
$$107$$ 2.56206e9i 1.88957i 0.327695 + 0.944784i $$0.393728\pi$$
−0.327695 + 0.944784i $$0.606272\pi$$
$$108$$ 0 0
$$109$$ − 4.72077e8i − 0.320327i −0.987090 0.160164i $$-0.948798\pi$$
0.987090 0.160164i $$-0.0512022\pi$$
$$110$$ 0 0
$$111$$ −1.73746e9 −1.08633
$$112$$ 0 0
$$113$$ 9.44729e8 0.545073 0.272536 0.962146i $$-0.412138\pi$$
0.272536 + 0.962146i $$0.412138\pi$$
$$114$$ 0 0
$$115$$ − 1.78779e9i − 0.953180i
$$116$$ 0 0
$$117$$ 1.85568e8i 0.0915517i
$$118$$ 0 0
$$119$$ −1.47931e9 −0.676237
$$120$$ 0 0
$$121$$ −1.42372e9 −0.603796
$$122$$ 0 0
$$123$$ 2.55699e9i 1.00729i
$$124$$ 0 0
$$125$$ − 1.82644e9i − 0.669132i
$$126$$ 0 0
$$127$$ 2.56290e9 0.874209 0.437105 0.899411i $$-0.356004\pi$$
0.437105 + 0.899411i $$0.356004\pi$$
$$128$$ 0 0
$$129$$ −7.63642e8 −0.242793
$$130$$ 0 0
$$131$$ 1.68489e9i 0.499863i 0.968264 + 0.249931i $$0.0804081\pi$$
−0.968264 + 0.249931i $$0.919592\pi$$
$$132$$ 0 0
$$133$$ − 2.80936e9i − 0.778531i
$$134$$ 0 0
$$135$$ −1.15761e9 −0.299957
$$136$$ 0 0
$$137$$ 5.71074e7 0.0138500 0.00692501 0.999976i $$-0.497796\pi$$
0.00692501 + 0.999976i $$0.497796\pi$$
$$138$$ 0 0
$$139$$ 5.87092e9i 1.33395i 0.745081 + 0.666974i $$0.232411\pi$$
−0.745081 + 0.666974i $$0.767589\pi$$
$$140$$ 0 0
$$141$$ − 2.91269e9i − 0.620594i
$$142$$ 0 0
$$143$$ −1.73930e9 −0.347826
$$144$$ 0 0
$$145$$ −9.67357e9 −1.81732
$$146$$ 0 0
$$147$$ − 1.48180e9i − 0.261734i
$$148$$ 0 0
$$149$$ 1.06329e10i 1.76731i 0.468136 + 0.883656i $$0.344926\pi$$
−0.468136 + 0.883656i $$0.655074\pi$$
$$150$$ 0 0
$$151$$ 4.05058e9 0.634046 0.317023 0.948418i $$-0.397317\pi$$
0.317023 + 0.948418i $$0.397317\pi$$
$$152$$ 0 0
$$153$$ −1.26738e9 −0.186980
$$154$$ 0 0
$$155$$ 5.66303e9i 0.788054i
$$156$$ 0 0
$$157$$ 5.91477e8i 0.0776944i 0.999245 + 0.0388472i $$0.0123686\pi$$
−0.999245 + 0.0388472i $$0.987631\pi$$
$$158$$ 0 0
$$159$$ −3.21728e9 −0.399210
$$160$$ 0 0
$$161$$ 6.28541e9 0.737254
$$162$$ 0 0
$$163$$ − 1.71216e10i − 1.89977i −0.312598 0.949886i $$-0.601199\pi$$
0.312598 0.949886i $$-0.398801\pi$$
$$164$$ 0 0
$$165$$ − 1.08501e10i − 1.13961i
$$166$$ 0 0
$$167$$ 1.79932e10 1.79013 0.895065 0.445936i $$-0.147129\pi$$
0.895065 + 0.445936i $$0.147129\pi$$
$$168$$ 0 0
$$169$$ 9.80454e9 0.924565
$$170$$ 0 0
$$171$$ − 2.40688e9i − 0.215264i
$$172$$ 0 0
$$173$$ 1.65957e10i 1.40861i 0.709900 + 0.704303i $$0.248740\pi$$
−0.709900 + 0.704303i $$0.751260\pi$$
$$174$$ 0 0
$$175$$ 2.13787e10 1.72310
$$176$$ 0 0
$$177$$ −7.58394e9 −0.580785
$$178$$ 0 0
$$179$$ 1.76363e10i 1.28401i 0.766701 + 0.642004i $$0.221897\pi$$
−0.766701 + 0.642004i $$0.778103\pi$$
$$180$$ 0 0
$$181$$ 4.78894e8i 0.0331654i 0.999862 + 0.0165827i $$0.00527868\pi$$
−0.999862 + 0.0165827i $$0.994721\pi$$
$$182$$ 0 0
$$183$$ 3.12976e9 0.206292
$$184$$ 0 0
$$185$$ 4.67235e10 2.93267
$$186$$ 0 0
$$187$$ − 1.18789e10i − 0.710380i
$$188$$ 0 0
$$189$$ − 4.06986e9i − 0.232007i
$$190$$ 0 0
$$191$$ 2.02911e10 1.10320 0.551601 0.834108i $$-0.314017\pi$$
0.551601 + 0.834108i $$0.314017\pi$$
$$192$$ 0 0
$$193$$ 6.29052e9 0.326346 0.163173 0.986597i $$-0.447827\pi$$
0.163173 + 0.986597i $$0.447827\pi$$
$$194$$ 0 0
$$195$$ − 4.99027e9i − 0.247154i
$$196$$ 0 0
$$197$$ 6.17930e9i 0.292308i 0.989262 + 0.146154i $$0.0466896\pi$$
−0.989262 + 0.146154i $$0.953310\pi$$
$$198$$ 0 0
$$199$$ −1.98276e10 −0.896255 −0.448128 0.893970i $$-0.647909\pi$$
−0.448128 + 0.893970i $$0.647909\pi$$
$$200$$ 0 0
$$201$$ −1.45908e10 −0.630516
$$202$$ 0 0
$$203$$ − 3.40099e10i − 1.40564i
$$204$$ 0 0
$$205$$ − 6.87623e10i − 2.71930i
$$206$$ 0 0
$$207$$ 5.38492e9 0.203851
$$208$$ 0 0
$$209$$ 2.25593e10 0.817838
$$210$$ 0 0
$$211$$ 5.37775e10i 1.86780i 0.357539 + 0.933898i $$0.383616\pi$$
−0.357539 + 0.933898i $$0.616384\pi$$
$$212$$ 0 0
$$213$$ 1.24893e10i 0.415748i
$$214$$ 0 0
$$215$$ 2.05358e10 0.655448
$$216$$ 0 0
$$217$$ −1.99098e10 −0.609534
$$218$$ 0 0
$$219$$ 3.07420e10i 0.903097i
$$220$$ 0 0
$$221$$ − 5.46347e9i − 0.154065i
$$222$$ 0 0
$$223$$ 4.21444e10 1.14122 0.570609 0.821222i $$-0.306707\pi$$
0.570609 + 0.821222i $$0.306707\pi$$
$$224$$ 0 0
$$225$$ 1.83158e10 0.476436
$$226$$ 0 0
$$227$$ 6.21393e10i 1.55328i 0.629944 + 0.776641i $$0.283078\pi$$
−0.629944 + 0.776641i $$0.716922\pi$$
$$228$$ 0 0
$$229$$ 2.28989e10i 0.550244i 0.961409 + 0.275122i $$0.0887182\pi$$
−0.961409 + 0.275122i $$0.911282\pi$$
$$230$$ 0 0
$$231$$ 3.81462e10 0.881449
$$232$$ 0 0
$$233$$ −6.44391e10 −1.43234 −0.716172 0.697923i $$-0.754108\pi$$
−0.716172 + 0.697923i $$0.754108\pi$$
$$234$$ 0 0
$$235$$ 7.83277e10i 1.67537i
$$236$$ 0 0
$$237$$ − 4.22227e10i − 0.869318i
$$238$$ 0 0
$$239$$ −1.85246e10 −0.367246 −0.183623 0.982997i $$-0.558783\pi$$
−0.183623 + 0.982997i $$0.558783\pi$$
$$240$$ 0 0
$$241$$ −8.59459e10 −1.64115 −0.820575 0.571539i $$-0.806347\pi$$
−0.820575 + 0.571539i $$0.806347\pi$$
$$242$$ 0 0
$$243$$ − 3.48678e9i − 0.0641500i
$$244$$ 0 0
$$245$$ 3.98483e10i 0.706581i
$$246$$ 0 0
$$247$$ 1.03757e10 0.177370
$$248$$ 0 0
$$249$$ −4.80357e10 −0.791894
$$250$$ 0 0
$$251$$ − 3.84815e10i − 0.611957i −0.952038 0.305978i $$-0.901016\pi$$
0.952038 0.305978i $$-0.0989835\pi$$
$$252$$ 0 0
$$253$$ 5.04720e10i 0.774477i
$$254$$ 0 0
$$255$$ 3.40822e10 0.504774
$$256$$ 0 0
$$257$$ 1.23429e11 1.76489 0.882447 0.470412i $$-0.155895\pi$$
0.882447 + 0.470412i $$0.155895\pi$$
$$258$$ 0 0
$$259$$ 1.64268e11i 2.26832i
$$260$$ 0 0
$$261$$ − 2.91374e10i − 0.388659i
$$262$$ 0 0
$$263$$ −2.36808e10 −0.305207 −0.152604 0.988287i $$-0.548766\pi$$
−0.152604 + 0.988287i $$0.548766\pi$$
$$264$$ 0 0
$$265$$ 8.65188e10 1.07771
$$266$$ 0 0
$$267$$ − 8.57204e9i − 0.103225i
$$268$$ 0 0
$$269$$ − 5.33379e10i − 0.621085i −0.950560 0.310542i $$-0.899489\pi$$
0.950560 0.310542i $$-0.100511\pi$$
$$270$$ 0 0
$$271$$ −1.06039e11 −1.19427 −0.597137 0.802139i $$-0.703695\pi$$
−0.597137 + 0.802139i $$0.703695\pi$$
$$272$$ 0 0
$$273$$ 1.75445e10 0.191166
$$274$$ 0 0
$$275$$ 1.71671e11i 1.81009i
$$276$$ 0 0
$$277$$ − 1.39984e10i − 0.142863i −0.997446 0.0714314i $$-0.977243\pi$$
0.997446 0.0714314i $$-0.0227567\pi$$
$$278$$ 0 0
$$279$$ −1.70574e10 −0.168536
$$280$$ 0 0
$$281$$ −9.66573e10 −0.924818 −0.462409 0.886667i $$-0.653015\pi$$
−0.462409 + 0.886667i $$0.653015\pi$$
$$282$$ 0 0
$$283$$ 1.62302e11i 1.50413i 0.659087 + 0.752066i $$0.270943\pi$$
−0.659087 + 0.752066i $$0.729057\pi$$
$$284$$ 0 0
$$285$$ 6.47255e10i 0.581130i
$$286$$ 0 0
$$287$$ 2.41751e11 2.10329
$$288$$ 0 0
$$289$$ −8.12738e10 −0.685347
$$290$$ 0 0
$$291$$ − 2.35914e10i − 0.192857i
$$292$$ 0 0
$$293$$ − 1.22365e10i − 0.0969957i −0.998823 0.0484979i $$-0.984557\pi$$
0.998823 0.0484979i $$-0.0154434\pi$$
$$294$$ 0 0
$$295$$ 2.03947e11 1.56790
$$296$$ 0 0
$$297$$ 3.26811e10 0.243721
$$298$$ 0 0
$$299$$ 2.32136e10i 0.167966i
$$300$$ 0 0
$$301$$ 7.21987e10i 0.506967i
$$302$$ 0 0
$$303$$ 1.11983e11 0.763237
$$304$$ 0 0
$$305$$ −8.41653e10 −0.556908
$$306$$ 0 0
$$307$$ − 1.03830e11i − 0.667116i −0.942730 0.333558i $$-0.891751\pi$$
0.942730 0.333558i $$-0.108249\pi$$
$$308$$ 0 0
$$309$$ 4.23001e10i 0.263954i
$$310$$ 0 0
$$311$$ 1.95717e10 0.118633 0.0593167 0.998239i $$-0.481108\pi$$
0.0593167 + 0.998239i $$0.481108\pi$$
$$312$$ 0 0
$$313$$ 2.36501e11 1.39278 0.696391 0.717662i $$-0.254788\pi$$
0.696391 + 0.717662i $$0.254788\pi$$
$$314$$ 0 0
$$315$$ 1.09446e11i 0.626330i
$$316$$ 0 0
$$317$$ 1.54506e11i 0.859369i 0.902979 + 0.429684i $$0.141375\pi$$
−0.902979 + 0.429684i $$0.858625\pi$$
$$318$$ 0 0
$$319$$ 2.73100e11 1.47660
$$320$$ 0 0
$$321$$ 2.07527e11 1.09094
$$322$$ 0 0
$$323$$ 7.08631e10i 0.362250i
$$324$$ 0 0
$$325$$ 7.89567e10i 0.392567i
$$326$$ 0 0
$$327$$ −3.82382e10 −0.184941
$$328$$ 0 0
$$329$$ −2.75380e11 −1.29584
$$330$$ 0 0
$$331$$ 4.67240e10i 0.213951i 0.994262 + 0.106975i $$0.0341166\pi$$
−0.994262 + 0.106975i $$0.965883\pi$$
$$332$$ 0 0
$$333$$ 1.40734e11i 0.627191i
$$334$$ 0 0
$$335$$ 3.92373e11 1.70215
$$336$$ 0 0
$$337$$ −4.64738e10 −0.196279 −0.0981394 0.995173i $$-0.531289\pi$$
−0.0981394 + 0.995173i $$0.531289\pi$$
$$338$$ 0 0
$$339$$ − 7.65231e10i − 0.314698i
$$340$$ 0 0
$$341$$ − 1.59876e11i − 0.640309i
$$342$$ 0 0
$$343$$ 1.68938e11 0.659027
$$344$$ 0 0
$$345$$ −1.44811e11 −0.550319
$$346$$ 0 0
$$347$$ 1.67034e11i 0.618476i 0.950985 + 0.309238i $$0.100074\pi$$
−0.950985 + 0.309238i $$0.899926\pi$$
$$348$$ 0 0
$$349$$ 4.14205e11i 1.49452i 0.664534 + 0.747258i $$0.268630\pi$$
−0.664534 + 0.747258i $$0.731370\pi$$
$$350$$ 0 0
$$351$$ 1.50310e10 0.0528574
$$352$$ 0 0
$$353$$ −3.92533e11 −1.34552 −0.672759 0.739862i $$-0.734891\pi$$
−0.672759 + 0.739862i $$0.734891\pi$$
$$354$$ 0 0
$$355$$ − 3.35861e11i − 1.12236i
$$356$$ 0 0
$$357$$ 1.19824e11i 0.390426i
$$358$$ 0 0
$$359$$ 5.24918e11 1.66789 0.833943 0.551850i $$-0.186078\pi$$
0.833943 + 0.551850i $$0.186078\pi$$
$$360$$ 0 0
$$361$$ 1.88112e11 0.582953
$$362$$ 0 0
$$363$$ 1.15321e11i 0.348602i
$$364$$ 0 0
$$365$$ − 8.26712e11i − 2.43801i
$$366$$ 0 0
$$367$$ 2.29807e11 0.661250 0.330625 0.943762i $$-0.392740\pi$$
0.330625 + 0.943762i $$0.392740\pi$$
$$368$$ 0 0
$$369$$ 2.07116e11 0.581561
$$370$$ 0 0
$$371$$ 3.04178e11i 0.833577i
$$372$$ 0 0
$$373$$ 5.03584e10i 0.134704i 0.997729 + 0.0673522i $$0.0214551\pi$$
−0.997729 + 0.0673522i $$0.978545\pi$$
$$374$$ 0 0
$$375$$ −1.47942e11 −0.386323
$$376$$ 0 0
$$377$$ 1.25607e11 0.320241
$$378$$ 0 0
$$379$$ 4.42233e11i 1.10097i 0.834846 + 0.550484i $$0.185557\pi$$
−0.834846 + 0.550484i $$0.814443\pi$$
$$380$$ 0 0
$$381$$ − 2.07595e11i − 0.504725i
$$382$$ 0 0
$$383$$ 3.96927e11 0.942576 0.471288 0.881979i $$-0.343789\pi$$
0.471288 + 0.881979i $$0.343789\pi$$
$$384$$ 0 0
$$385$$ −1.02582e12 −2.37957
$$386$$ 0 0
$$387$$ 6.18550e10i 0.140177i
$$388$$ 0 0
$$389$$ − 7.68232e11i − 1.70106i −0.525929 0.850529i $$-0.676282\pi$$
0.525929 0.850529i $$-0.323718\pi$$
$$390$$ 0 0
$$391$$ −1.58542e11 −0.343044
$$392$$ 0 0
$$393$$ 1.36476e11 0.288596
$$394$$ 0 0
$$395$$ 1.13545e12i 2.34682i
$$396$$ 0 0
$$397$$ − 3.41102e11i − 0.689172i −0.938755 0.344586i $$-0.888019\pi$$
0.938755 0.344586i $$-0.111981\pi$$
$$398$$ 0 0
$$399$$ −2.27559e11 −0.449485
$$400$$ 0 0
$$401$$ 5.83691e11 1.12728 0.563642 0.826019i $$-0.309400\pi$$
0.563642 + 0.826019i $$0.309400\pi$$
$$402$$ 0 0
$$403$$ − 7.35318e10i − 0.138868i
$$404$$ 0 0
$$405$$ 9.37662e10i 0.173180i
$$406$$ 0 0
$$407$$ −1.31908e12 −2.38284
$$408$$ 0 0
$$409$$ 7.80713e11 1.37955 0.689773 0.724026i $$-0.257710\pi$$
0.689773 + 0.724026i $$0.257710\pi$$
$$410$$ 0 0
$$411$$ − 4.62570e9i − 0.00799631i
$$412$$ 0 0
$$413$$ 7.17025e11i 1.21272i
$$414$$ 0 0
$$415$$ 1.29177e12 2.13781
$$416$$ 0 0
$$417$$ 4.75544e11 0.770156
$$418$$ 0 0
$$419$$ − 5.88488e11i − 0.932769i −0.884582 0.466385i $$-0.845556\pi$$
0.884582 0.466385i $$-0.154444\pi$$
$$420$$ 0 0
$$421$$ − 2.50950e11i − 0.389330i −0.980870 0.194665i $$-0.937638\pi$$
0.980870 0.194665i $$-0.0623620\pi$$
$$422$$ 0 0
$$423$$ −2.35928e11 −0.358300
$$424$$ 0 0
$$425$$ −5.39253e11 −0.801756
$$426$$ 0 0
$$427$$ − 2.95904e11i − 0.430750i
$$428$$ 0 0
$$429$$ 1.40883e11i 0.200817i
$$430$$ 0 0
$$431$$ −1.22079e12 −1.70409 −0.852046 0.523467i $$-0.824638\pi$$
−0.852046 + 0.523467i $$0.824638\pi$$
$$432$$ 0 0
$$433$$ 2.85938e11 0.390909 0.195455 0.980713i $$-0.437382\pi$$
0.195455 + 0.980713i $$0.437382\pi$$
$$434$$ 0 0
$$435$$ 7.83559e11i 1.04923i
$$436$$ 0 0
$$437$$ − 3.01088e11i − 0.394936i
$$438$$ 0 0
$$439$$ −4.26567e11 −0.548147 −0.274073 0.961709i $$-0.588371\pi$$
−0.274073 + 0.961709i $$0.588371\pi$$
$$440$$ 0 0
$$441$$ −1.20025e11 −0.151112
$$442$$ 0 0
$$443$$ − 2.78131e11i − 0.343109i −0.985175 0.171555i $$-0.945121\pi$$
0.985175 0.171555i $$-0.0548790\pi$$
$$444$$ 0 0
$$445$$ 2.30518e11i 0.278667i
$$446$$ 0 0
$$447$$ 8.61265e11 1.02036
$$448$$ 0 0
$$449$$ 6.47195e11 0.751496 0.375748 0.926722i $$-0.377386\pi$$
0.375748 + 0.926722i $$0.377386\pi$$
$$450$$ 0 0
$$451$$ 1.94127e12i 2.20948i
$$452$$ 0 0
$$453$$ − 3.28097e11i − 0.366067i
$$454$$ 0 0
$$455$$ −4.71806e11 −0.516074
$$456$$ 0 0
$$457$$ 9.87836e11 1.05940 0.529702 0.848184i $$-0.322304\pi$$
0.529702 + 0.848184i $$0.322304\pi$$
$$458$$ 0 0
$$459$$ 1.02658e11i 0.107953i
$$460$$ 0 0
$$461$$ 5.23286e11i 0.539616i 0.962914 + 0.269808i $$0.0869603\pi$$
−0.962914 + 0.269808i $$0.913040\pi$$
$$462$$ 0 0
$$463$$ −9.21613e11 −0.932039 −0.466020 0.884774i $$-0.654312\pi$$
−0.466020 + 0.884774i $$0.654312\pi$$
$$464$$ 0 0
$$465$$ 4.58705e11 0.454983
$$466$$ 0 0
$$467$$ − 1.04570e11i − 0.101738i −0.998705 0.0508689i $$-0.983801\pi$$
0.998705 0.0508689i $$-0.0161991\pi$$
$$468$$ 0 0
$$469$$ 1.37949e12i 1.31656i
$$470$$ 0 0
$$471$$ 4.79097e10 0.0448569
$$472$$ 0 0
$$473$$ −5.79758e11 −0.532563
$$474$$ 0 0
$$475$$ − 1.02409e12i − 0.923036i
$$476$$ 0 0
$$477$$ 2.60600e11i 0.230484i
$$478$$ 0 0
$$479$$ 1.93040e12 1.67547 0.837737 0.546074i $$-0.183878\pi$$
0.837737 + 0.546074i $$0.183878\pi$$
$$480$$ 0 0
$$481$$ −6.06683e11 −0.516784
$$482$$ 0 0
$$483$$ − 5.09118e11i − 0.425654i
$$484$$ 0 0
$$485$$ 6.34417e11i 0.520639i
$$486$$ 0 0
$$487$$ −1.28669e12 −1.03656 −0.518281 0.855210i $$-0.673428\pi$$
−0.518281 + 0.855210i $$0.673428\pi$$
$$488$$ 0 0
$$489$$ −1.38685e12 −1.09683
$$490$$ 0 0
$$491$$ − 1.58967e12i − 1.23436i −0.786823 0.617179i $$-0.788276\pi$$
0.786823 0.617179i $$-0.211724\pi$$
$$492$$ 0 0
$$493$$ 8.57860e11i 0.654042i
$$494$$ 0 0
$$495$$ −8.78857e11 −0.657953
$$496$$ 0 0
$$497$$ 1.18080e12 0.868109
$$498$$ 0 0
$$499$$ − 1.27745e12i − 0.922344i −0.887311 0.461172i $$-0.847429\pi$$
0.887311 0.461172i $$-0.152571\pi$$
$$500$$ 0 0
$$501$$ − 1.45745e12i − 1.03353i
$$502$$ 0 0
$$503$$ −2.40613e11 −0.167596 −0.0837979 0.996483i $$-0.526705\pi$$
−0.0837979 + 0.996483i $$0.526705\pi$$
$$504$$ 0 0
$$505$$ −3.01143e12 −2.06045
$$506$$ 0 0
$$507$$ − 7.94168e11i − 0.533798i
$$508$$ 0 0
$$509$$ − 5.46994e11i − 0.361204i −0.983556 0.180602i $$-0.942195\pi$$
0.983556 0.180602i $$-0.0578046\pi$$
$$510$$ 0 0
$$511$$ 2.90651e12 1.88572
$$512$$ 0 0
$$513$$ −1.94957e11 −0.124283
$$514$$ 0 0
$$515$$ − 1.13753e12i − 0.712574i
$$516$$ 0 0
$$517$$ − 2.21131e12i − 1.36127i
$$518$$ 0 0
$$519$$ 1.34426e12 0.813259
$$520$$ 0 0
$$521$$ 2.49243e12 1.48202 0.741009 0.671495i $$-0.234347\pi$$
0.741009 + 0.671495i $$0.234347\pi$$
$$522$$ 0 0
$$523$$ − 1.54454e12i − 0.902697i −0.892348 0.451349i $$-0.850943\pi$$
0.892348 0.451349i $$-0.149057\pi$$
$$524$$ 0 0
$$525$$ − 1.73167e12i − 0.994829i
$$526$$ 0 0
$$527$$ 5.02202e11 0.283616
$$528$$ 0 0
$$529$$ −1.12753e12 −0.626004
$$530$$ 0 0
$$531$$ 6.14299e11i 0.335316i
$$532$$ 0 0
$$533$$ 8.92846e11i 0.479186i
$$534$$ 0 0
$$535$$ −5.58079e12 −2.94513
$$536$$ 0 0
$$537$$ 1.42854e12 0.741322
$$538$$ 0 0
$$539$$ − 1.12498e12i − 0.574110i
$$540$$ 0 0
$$541$$ − 4.66468e11i − 0.234118i −0.993125 0.117059i $$-0.962653\pi$$
0.993125 0.117059i $$-0.0373466\pi$$
$$542$$ 0 0
$$543$$ 3.87904e10 0.0191481
$$544$$ 0 0
$$545$$ 1.02830e12 0.499269
$$546$$ 0 0
$$547$$ 1.63835e12i 0.782462i 0.920293 + 0.391231i $$0.127951\pi$$
−0.920293 + 0.391231i $$0.872049\pi$$
$$548$$ 0 0
$$549$$ − 2.53511e11i − 0.119103i
$$550$$ 0 0
$$551$$ −1.62916e12 −0.752978
$$552$$ 0 0
$$553$$ −3.99195e12 −1.81519
$$554$$ 0 0
$$555$$ − 3.78460e12i − 1.69318i
$$556$$ 0 0
$$557$$ 6.77404e11i 0.298194i 0.988823 + 0.149097i $$0.0476367\pi$$
−0.988823 + 0.149097i $$0.952363\pi$$
$$558$$ 0 0
$$559$$ −2.66647e11 −0.115501
$$560$$ 0 0
$$561$$ −9.62194e11 −0.410138
$$562$$ 0 0
$$563$$ − 2.75360e12i − 1.15508i −0.816361 0.577542i $$-0.804012\pi$$
0.816361 0.577542i $$-0.195988\pi$$
$$564$$ 0 0
$$565$$ 2.05785e12i 0.849563i
$$566$$ 0 0
$$567$$ −3.29659e11 −0.133949
$$568$$ 0 0
$$569$$ 2.14461e12 0.857716 0.428858 0.903372i $$-0.358916\pi$$
0.428858 + 0.903372i $$0.358916\pi$$
$$570$$ 0 0
$$571$$ − 3.43875e12i − 1.35375i −0.736098 0.676875i $$-0.763334\pi$$
0.736098 0.676875i $$-0.236666\pi$$
$$572$$ 0 0
$$573$$ − 1.64358e12i − 0.636934i
$$574$$ 0 0
$$575$$ 2.29121e12 0.874098
$$576$$ 0 0
$$577$$ −2.85300e12 −1.07155 −0.535773 0.844362i $$-0.679980\pi$$
−0.535773 + 0.844362i $$0.679980\pi$$
$$578$$ 0 0
$$579$$ − 5.09532e11i − 0.188416i
$$580$$ 0 0
$$581$$ 4.54154e12i 1.65353i
$$582$$ 0 0
$$583$$ −2.44256e12 −0.875663
$$584$$ 0 0
$$585$$ −4.04212e11 −0.142695
$$586$$ 0 0
$$587$$ − 2.36076e10i − 0.00820691i −0.999992 0.00410346i $$-0.998694\pi$$
0.999992 0.00410346i $$-0.00130617\pi$$
$$588$$ 0 0
$$589$$ 9.53732e11i 0.326518i
$$590$$ 0 0
$$591$$ 5.00523e11 0.168764
$$592$$ 0 0
$$593$$ −2.05586e12 −0.682727 −0.341364 0.939931i $$-0.610889\pi$$
−0.341364 + 0.939931i $$0.610889\pi$$
$$594$$ 0 0
$$595$$ − 3.22231e12i − 1.05400i
$$596$$ 0 0
$$597$$ 1.60604e12i 0.517453i
$$598$$ 0 0
$$599$$ −5.72088e12 −1.81569 −0.907846 0.419304i $$-0.862274\pi$$
−0.907846 + 0.419304i $$0.862274\pi$$
$$600$$ 0 0
$$601$$ 3.10411e12 0.970516 0.485258 0.874371i $$-0.338726\pi$$
0.485258 + 0.874371i $$0.338726\pi$$
$$602$$ 0 0
$$603$$ 1.18185e12i 0.364028i
$$604$$ 0 0
$$605$$ − 3.10121e12i − 0.941091i
$$606$$ 0 0
$$607$$ −1.37507e12 −0.411127 −0.205564 0.978644i $$-0.565903\pi$$
−0.205564 + 0.978644i $$0.565903\pi$$
$$608$$ 0 0
$$609$$ −2.75480e12 −0.811544
$$610$$ 0 0
$$611$$ − 1.01705e12i − 0.295227i
$$612$$ 0 0
$$613$$ 6.69362e12i 1.91465i 0.289016 + 0.957324i $$0.406672\pi$$
−0.289016 + 0.957324i $$0.593328\pi$$
$$614$$ 0 0
$$615$$ −5.56974e12 −1.56999
$$616$$ 0 0
$$617$$ 4.17726e11 0.116040 0.0580201 0.998315i $$-0.481521\pi$$
0.0580201 + 0.998315i $$0.481521\pi$$
$$618$$ 0 0
$$619$$ 6.34992e12i 1.73844i 0.494423 + 0.869221i $$0.335379\pi$$
−0.494423 + 0.869221i $$0.664621\pi$$
$$620$$ 0 0
$$621$$ − 4.36178e11i − 0.117693i
$$622$$ 0 0
$$623$$ −8.10445e11 −0.215540
$$624$$ 0 0
$$625$$ −1.47394e12 −0.386385
$$626$$ 0 0
$$627$$ − 1.82730e12i − 0.472179i
$$628$$ 0 0
$$629$$ − 4.14348e12i − 1.05545i
$$630$$ 0 0
$$631$$ −4.73127e12 −1.18808 −0.594040 0.804435i $$-0.702468\pi$$
−0.594040 + 0.804435i $$0.702468\pi$$
$$632$$ 0 0
$$633$$ 4.35598e12 1.07837
$$634$$ 0 0
$$635$$ 5.58263e12i 1.36256i
$$636$$ 0 0
$$637$$ − 5.17411e11i − 0.124511i
$$638$$ 0 0
$$639$$ 1.01163e12 0.240032
$$640$$ 0 0
$$641$$ 3.90673e12 0.914014 0.457007 0.889463i $$-0.348921\pi$$
0.457007 + 0.889463i $$0.348921\pi$$
$$642$$ 0 0
$$643$$ − 2.06347e12i − 0.476046i −0.971260 0.238023i $$-0.923501\pi$$
0.971260 0.238023i $$-0.0764993\pi$$
$$644$$ 0 0
$$645$$ − 1.66340e12i − 0.378423i
$$646$$ 0 0
$$647$$ −4.14405e12 −0.929728 −0.464864 0.885382i $$-0.653897\pi$$
−0.464864 + 0.885382i $$0.653897\pi$$
$$648$$ 0 0
$$649$$ −5.75773e12 −1.27394
$$650$$ 0 0
$$651$$ 1.61269e12i 0.351915i
$$652$$ 0 0
$$653$$ 8.11599e12i 1.74676i 0.487044 + 0.873378i $$0.338075\pi$$
−0.487044 + 0.873378i $$0.661925\pi$$
$$654$$ 0 0
$$655$$ −3.67010e12 −0.779098
$$656$$ 0 0
$$657$$ 2.49011e12 0.521403
$$658$$ 0 0
$$659$$ 8.57180e11i 0.177047i 0.996074 + 0.0885234i $$0.0282148\pi$$
−0.996074 + 0.0885234i $$0.971785\pi$$
$$660$$ 0 0
$$661$$ − 1.57325e12i − 0.320546i −0.987073 0.160273i $$-0.948762\pi$$
0.987073 0.160273i $$-0.0512375\pi$$
$$662$$ 0 0
$$663$$ −4.42541e11 −0.0889494
$$664$$ 0 0
$$665$$ 6.11948e12 1.21344
$$666$$ 0 0
$$667$$ − 3.64493e12i − 0.713055i
$$668$$ 0 0
$$669$$ − 3.41370e12i − 0.658882i
$$670$$ 0 0
$$671$$ 2.37612e12 0.452498
$$672$$ 0 0
$$673$$ 4.91142e12 0.922867 0.461434 0.887175i $$-0.347335\pi$$
0.461434 + 0.887175i $$0.347335\pi$$
$$674$$ 0 0
$$675$$ − 1.48358e12i − 0.275071i
$$676$$ 0 0
$$677$$ − 4.67942e12i − 0.856137i −0.903746 0.428069i $$-0.859194\pi$$
0.903746 0.428069i $$-0.140806\pi$$
$$678$$ 0 0
$$679$$ −2.23045e12 −0.402697
$$680$$ 0 0
$$681$$ 5.03329e12 0.896787
$$682$$ 0 0
$$683$$ − 5.92535e12i − 1.04189i −0.853591 0.520944i $$-0.825580\pi$$
0.853591 0.520944i $$-0.174420\pi$$
$$684$$ 0 0
$$685$$ 1.24394e11i 0.0215870i
$$686$$ 0 0
$$687$$ 1.85481e12 0.317683
$$688$$ 0 0
$$689$$ −1.12341e12 −0.189911
$$690$$ 0 0
$$691$$ − 4.43776e12i − 0.740479i −0.928936 0.370239i $$-0.879276\pi$$
0.928936 0.370239i $$-0.120724\pi$$
$$692$$ 0 0
$$693$$ − 3.08984e12i − 0.508905i
$$694$$ 0 0
$$695$$ −1.27883e13 −2.07912
$$696$$ 0 0
$$697$$ −6.09789e12 −0.978661
$$698$$ 0 0
$$699$$ 5.21956e12i 0.826965i
$$700$$ 0 0
$$701$$ − 9.55720e12i − 1.49486i −0.664343 0.747428i $$-0.731288\pi$$
0.664343 0.747428i $$-0.268712\pi$$
$$702$$ 0 0
$$703$$ 7.86888e12 1.21511
$$704$$ 0 0
$$705$$ 6.34454e12 0.967273
$$706$$ 0 0
$$707$$ − 1.05874e13i − 1.59369i
$$708$$ 0 0
$$709$$ 5.67373e12i 0.843258i 0.906769 + 0.421629i $$0.138541\pi$$
−0.906769 + 0.421629i $$0.861459\pi$$
$$710$$ 0 0
$$711$$ −3.42004e12 −0.501901
$$712$$ 0 0
$$713$$ −2.13379e12 −0.309207
$$714$$ 0 0
$$715$$ − 3.78862e12i − 0.542130i
$$716$$ 0 0
$$717$$ 1.50049e12i 0.212030i
$$718$$ 0 0
$$719$$ 3.80229e11 0.0530598 0.0265299 0.999648i $$-0.491554\pi$$
0.0265299 + 0.999648i $$0.491554\pi$$
$$720$$ 0 0
$$721$$ 3.99927e12 0.551152
$$722$$ 0 0
$$723$$ 6.96161e12i 0.947518i
$$724$$ 0 0
$$725$$ − 1.23976e13i − 1.66654i
$$726$$ 0 0
$$727$$ 1.32242e13 1.75576 0.877880 0.478881i $$-0.158957\pi$$
0.877880 + 0.478881i $$0.158957\pi$$
$$728$$ 0 0
$$729$$ −2.82430e11 −0.0370370
$$730$$ 0 0
$$731$$ − 1.82113e12i − 0.235892i
$$732$$ 0 0
$$733$$ 1.53563e13i 1.96480i 0.186797 + 0.982399i $$0.440189\pi$$
−0.186797 + 0.982399i $$0.559811\pi$$
$$734$$ 0 0
$$735$$ 3.22771e12 0.407945
$$736$$ 0 0
$$737$$ −1.10773e13 −1.38303
$$738$$ 0 0
$$739$$ − 6.82404e12i − 0.841669i −0.907137 0.420835i $$-0.861737\pi$$
0.907137 0.420835i $$-0.138263\pi$$
$$740$$ 0 0
$$741$$ − 8.40430e11i − 0.102405i
$$742$$ 0 0
$$743$$ 6.28798e12 0.756940 0.378470 0.925614i $$-0.376450\pi$$
0.378470 + 0.925614i $$0.376450\pi$$
$$744$$ 0 0
$$745$$ −2.31610e13 −2.75458
$$746$$ 0 0
$$747$$ 3.89089e12i 0.457200i
$$748$$ 0 0
$$749$$ − 1.96207e13i − 2.27796i
$$750$$ 0 0
$$751$$ 1.19773e13 1.37397 0.686986 0.726671i $$-0.258933\pi$$
0.686986 + 0.726671i $$0.258933\pi$$
$$752$$ 0 0
$$753$$ −3.11700e12 −0.353313
$$754$$ 0 0
$$755$$ 8.82315e12i 0.988240i
$$756$$ 0 0
$$757$$ 5.00435e12i 0.553881i 0.960887 + 0.276940i $$0.0893205\pi$$
−0.960887 + 0.276940i $$0.910680\pi$$
$$758$$ 0 0
$$759$$ 4.08823e12 0.447144
$$760$$ 0 0
$$761$$ 2.44206e12 0.263952 0.131976 0.991253i $$-0.457868\pi$$
0.131976 + 0.991253i $$0.457868\pi$$
$$762$$ 0 0
$$763$$ 3.61524e12i 0.386168i
$$764$$ 0 0
$$765$$ − 2.76066e12i − 0.291431i
$$766$$ 0 0
$$767$$ −2.64815e12 −0.276289
$$768$$ 0 0
$$769$$ 8.52460e12 0.879034 0.439517 0.898234i $$-0.355150\pi$$
0.439517 + 0.898234i $$0.355150\pi$$
$$770$$ 0 0
$$771$$ − 9.99776e12i − 1.01896i
$$772$$ 0 0
$$773$$ − 5.84609e12i − 0.588922i −0.955664 0.294461i $$-0.904860\pi$$
0.955664 0.294461i $$-0.0951401\pi$$
$$774$$ 0 0
$$775$$ −7.25769e12 −0.722672
$$776$$ 0 0
$$777$$ 1.33057e13 1.30962
$$778$$ 0 0
$$779$$ − 1.15805e13i − 1.12670i
$$780$$ 0 0
$$781$$ 9.48190e12i 0.911939i
$$782$$ 0 0
$$783$$ −2.36013e12 −0.224392
$$784$$ 0 0
$$785$$ −1.28838e12 −0.121096
$$786$$ 0 0
$$787$$ − 1.32719e13i − 1.23324i −0.787262 0.616618i $$-0.788502\pi$$
0.787262 0.616618i $$-0.211498\pi$$
$$788$$ 0 0
$$789$$ 1.91814e12i 0.176211i
$$790$$ 0 0
$$791$$ −7.23489e12 −0.657109
$$792$$ 0 0
$$793$$ 1.09285e12 0.0981363
$$794$$ 0 0
$$795$$ − 7.00802e12i − 0.622219i
$$796$$ 0 0
$$797$$ − 2.02632e12i − 0.177887i −0.996037 0.0889436i $$-0.971651\pi$$
0.996037 0.0889436i $$-0.0283491\pi$$
$$798$$ 0 0
$$799$$ 6.94616e12 0.602954
$$800$$ 0 0
$$801$$ −6.94335e11 −0.0595968
$$802$$ 0 0
$$803$$ 2.33394e13i 1.98093i
$$804$$ 0 0
$$805$$ 1.36911e13i 1.14910i
$$806$$ 0 0
$$807$$ −4.32037e12 −0.358584
$$808$$ 0 0
$$809$$ −3.71860e12 −0.305219 −0.152609 0.988287i $$-0.548768\pi$$
−0.152609 + 0.988287i $$0.548768\pi$$
$$810$$ 0 0
$$811$$ 8.32783e12i 0.675987i 0.941149 + 0.337993i $$0.109748\pi$$
−0.941149 + 0.337993i $$0.890252\pi$$
$$812$$ 0 0
$$813$$ 8.58916e12i 0.689515i
$$814$$ 0 0
$$815$$ 3.72951e13 2.96103
$$816$$ 0 0
$$817$$ 3.45851e12 0.271575
$$818$$ 0 0
$$819$$ − 1.42111e12i − 0.110370i
$$820$$ 0 0
$$821$$ 3.72889e12i 0.286441i 0.989691 + 0.143220i $$0.0457458\pi$$
−0.989691 + 0.143220i $$0.954254\pi$$
$$822$$ 0 0
$$823$$ −4.95662e12 −0.376605 −0.188303 0.982111i $$-0.560299\pi$$
−0.188303 + 0.982111i $$0.560299\pi$$
$$824$$ 0 0
$$825$$ 1.39054e13 1.04506
$$826$$ 0 0
$$827$$ − 2.05532e12i − 0.152794i −0.997077 0.0763968i $$-0.975658\pi$$
0.997077 0.0763968i $$-0.0243416\pi$$
$$828$$ 0 0
$$829$$ − 4.49719e12i − 0.330709i −0.986234 0.165354i $$-0.947123\pi$$
0.986234 0.165354i $$-0.0528768\pi$$
$$830$$ 0 0
$$831$$ −1.13387e12 −0.0824818
$$832$$ 0 0
$$833$$ 3.53378e12 0.254294
$$834$$ 0 0
$$835$$ 3.91936e13i 2.79014i
$$836$$ 0 0
$$837$$ 1.38165e12i 0.0973045i
$$838$$ 0 0
$$839$$ 9.05888e12 0.631169 0.315585 0.948897i $$-0.397799\pi$$
0.315585 + 0.948897i $$0.397799\pi$$
$$840$$ 0 0
$$841$$ −5.21531e12 −0.359499
$$842$$ 0 0
$$843$$ 7.82924e12i 0.533944i
$$844$$ 0 0
$$845$$ 2.13567e13i 1.44105i
$$846$$ 0 0
$$847$$ 1.09031e13 0.727903
$$848$$ 0 0
$$849$$ 1.31465e13 0.868411
$$850$$ 0 0
$$851$$ 1.76051e13i 1.15068i
$$852$$ 0 0
$$853$$ 2.43668e13i 1.57589i 0.615743 + 0.787947i $$0.288856\pi$$
−0.615743 + 0.787947i $$0.711144\pi$$
$$854$$ 0 0
$$855$$ 5.24276e12 0.335516
$$856$$ 0 0
$$857$$ 1.54772e13 0.980117 0.490059 0.871689i $$-0.336975\pi$$
0.490059 + 0.871689i $$0.336975\pi$$
$$858$$ 0 0
$$859$$ 1.56032e12i 0.0977790i 0.998804 + 0.0488895i $$0.0155682\pi$$
−0.998804 + 0.0488895i $$0.984432\pi$$
$$860$$ 0 0
$$861$$ − 1.95818e13i − 1.21434i
$$862$$ 0 0
$$863$$ −1.09332e13 −0.670962 −0.335481 0.942047i $$-0.608899\pi$$
−0.335481 + 0.942047i $$0.608899\pi$$
$$864$$ 0 0
$$865$$ −3.61496e13 −2.19549
$$866$$ 0 0
$$867$$ 6.58318e12i 0.395685i
$$868$$ 0 0
$$869$$ − 3.20555e13i − 1.90684i
$$870$$ 0 0
$$871$$ −5.09478e12 −0.299947
$$872$$ 0 0
$$873$$ −1.91090e12 −0.111346
$$874$$ 0 0
$$875$$ 1.39872e13i 0.806668i
$$876$$ 0 0
$$877$$ 1.51832e13i 0.866693i 0.901227 + 0.433347i $$0.142667\pi$$
−0.901227 + 0.433347i $$0.857333\pi$$
$$878$$ 0 0
$$879$$ −9.91156e11 −0.0560005
$$880$$ 0 0
$$881$$ 2.87715e13 1.60905 0.804527 0.593916i $$-0.202419\pi$$
0.804527 + 0.593916i $$0.202419\pi$$
$$882$$ 0 0
$$883$$ 2.30013e13i 1.27329i 0.771155 + 0.636647i $$0.219679\pi$$
−0.771155 + 0.636647i $$0.780321\pi$$
$$884$$ 0 0
$$885$$ − 1.65197e13i − 0.905225i
$$886$$ 0 0
$$887$$ 1.27341e13 0.690735 0.345367 0.938468i $$-0.387754\pi$$
0.345367 + 0.938468i $$0.387754\pi$$
$$888$$ 0 0
$$889$$ −1.96271e13 −1.05390
$$890$$ 0 0
$$891$$ − 2.64717e12i − 0.140712i
$$892$$ 0 0
$$893$$ 1.31914e13i 0.694162i
$$894$$ 0 0
$$895$$ −3.84161e13 −2.00129
$$896$$ 0 0
$$897$$ 1.88030e12 0.0969752
$$898$$ 0 0
$$899$$ 1.15458e13i 0.589528i
$$900$$ 0 0
$$901$$ − 7.67255e12i − 0.387863i
$$902$$ 0 0
$$903$$ 5.84809e12 0.292698
$$904$$ 0 0
$$905$$ −1.04315e12 −0.0516924
$$906$$ 0 0
$$907$$ 2.44366e12i 0.119897i 0.998201 + 0.0599484i $$0.0190936\pi$$
−0.998201 + 0.0599484i $$0.980906\pi$$
$$908$$ 0 0
$$909$$ − 9.07061e12i − 0.440655i
$$910$$ 0 0
$$911$$ 3.69051e12 0.177523 0.0887613 0.996053i $$-0.471709\pi$$
0.0887613 + 0.996053i $$0.471709\pi$$
$$912$$ 0 0
$$913$$ −3.64688e13 −1.73701
$$914$$ 0 0
$$915$$ 6.81739e12i 0.321531i
$$916$$ 0 0
$$917$$ − 1.29032e13i − 0.602607i
$$918$$ 0 0
$$919$$ 1.04045e13 0.481173 0.240587 0.970628i $$-0.422660\pi$$
0.240587 + 0.970628i $$0.422660\pi$$
$$920$$ 0 0
$$921$$ −8.41025e12 −0.385160
$$922$$ 0 0
$$923$$ 4.36100e12i 0.197778i
$$924$$ 0 0
$$925$$ 5.98805e13i 2.68935i
$$926$$ 0 0
$$927$$ 3.42631e12 0.152394
$$928$$ 0 0
$$929$$ −3.78338e13 −1.66652 −0.833258 0.552885i $$-0.813527\pi$$
−0.833258 + 0.552885i $$0.813527\pi$$
$$930$$ 0 0
$$931$$ 6.71100e12i 0.292761i
$$932$$ 0 0
$$933$$ − 1.58531e12i − 0.0684930i
$$934$$ 0 0
$$935$$ 2.58752e13 1.10721
$$936$$ 0 0
$$937$$ −7.67680e12 −0.325351 −0.162675 0.986680i $$-0.552012\pi$$
−0.162675 + 0.986680i $$0.552012\pi$$
$$938$$ 0 0
$$939$$ − 1.91566e13i − 0.804123i
$$940$$ 0 0
$$941$$ − 1.98175e12i − 0.0823941i −0.999151 0.0411970i $$-0.986883\pi$$
0.999151 0.0411970i $$-0.0131171\pi$$
$$942$$ 0 0
$$943$$ 2.59091e13 1.06696
$$944$$ 0 0
$$945$$ 8.86514e12 0.361612
$$946$$ 0 0
$$947$$ 1.68460e13i 0.680647i 0.940308 + 0.340324i $$0.110537\pi$$
−0.940308 + 0.340324i $$0.889463\pi$$
$$948$$ 0 0
$$949$$ 1.07345e13i 0.429618i
$$950$$ 0 0
$$951$$ 1.25150e13 0.496157
$$952$$ 0 0
$$953$$ 2.24737e13 0.882585 0.441293 0.897363i $$-0.354520\pi$$
0.441293 + 0.897363i $$0.354520\pi$$
$$954$$ 0 0
$$955$$ 4.41989e13i 1.71948i
$$956$$ 0 0
$$957$$ − 2.21211e13i − 0.852518i
$$958$$ 0 0
$$959$$ −4.37338e11 −0.0166968
$$960$$ 0 0
$$961$$ −1.96806e13 −0.744359
$$962$$ 0 0
$$963$$ − 1.68097e13i − 0.629856i
$$964$$ 0 0
$$965$$ 1.37023e13i 0.508651i
$$966$$ 0 0
$$967$$ −3.69178e13 −1.35774 −0.678871 0.734258i $$-0.737530\pi$$
−0.678871 + 0.734258i $$0.737530\pi$$
$$968$$ 0 0
$$969$$ 5.73991e12 0.209145
$$970$$ 0 0
$$971$$ 1.73104e13i 0.624913i 0.949932 + 0.312456i $$0.101152\pi$$
−0.949932 + 0.312456i $$0.898848\pi$$
$$972$$ 0 0
$$973$$ − 4.49604e13i − 1.60813i
$$974$$ 0 0
$$975$$ 6.39549e12 0.226649
$$976$$ 0 0
$$977$$ 2.58899e13 0.909085 0.454543 0.890725i $$-0.349803\pi$$
0.454543 + 0.890725i $$0.349803\pi$$
$$978$$ 0 0
$$979$$ − 6.50790e12i − 0.226422i
$$980$$ 0 0
$$981$$ 3.09730e12i 0.106776i
$$982$$ 0 0
$$983$$ 2.22442e13 0.759848 0.379924 0.925018i $$-0.375950\pi$$
0.379924 + 0.925018i $$0.375950\pi$$
$$984$$ 0 0
$$985$$ −1.34600e13 −0.455599
$$986$$ 0 0
$$987$$ 2.23058e13i 0.748154i
$$988$$ 0 0
$$989$$ 7.73774e12i 0.257176i
$$990$$ 0 0
$$991$$ 3.85698e13 1.27033 0.635165 0.772377i $$-0.280932\pi$$
0.635165 + 0.772377i $$0.280932\pi$$
$$992$$ 0 0
$$993$$ 3.78465e12 0.123525
$$994$$ 0 0
$$995$$ − 4.31894e13i − 1.39693i
$$996$$ 0 0
$$997$$ − 1.29804e13i − 0.416062i −0.978122 0.208031i $$-0.933294\pi$$
0.978122 0.208031i $$-0.0667056\pi$$
$$998$$ 0 0
$$999$$ 1.13995e13 0.362109
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 384.10.d.a.193.4 8
4.3 odd 2 384.10.d.b.193.8 yes 8
8.3 odd 2 384.10.d.b.193.1 yes 8
8.5 even 2 inner 384.10.d.a.193.5 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.a.193.4 8 1.1 even 1 trivial
384.10.d.a.193.5 yes 8 8.5 even 2 inner
384.10.d.b.193.1 yes 8 8.3 odd 2
384.10.d.b.193.8 yes 8 4.3 odd 2